IJCFeedback control of linear discrete-time systems under state and control constraints88

8/10/2019 IJCFeedback control of linear discrete-time systems under state and control constraints88

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Feedback control of linear discrete-time systems under state and

control constraints

In this paper the problem of stabilizing linear discrete-time systems under state and

control linear constraints is studied. Based on the concept of positive invariance,

existence conditions of linear state feedback control laws respecting both the

constraints are established. These conditions are then translated into an algorithm

of linear programming.

1. Introduction

Most industrial systems must operate within fixed bounds and are subject to strict

control limitations. The determination of closed-loop controls for such systems by

state or output feedback often reduces to solving an associated unconstrained

problem and then modifying the solution by superimposition of state and control

limitations. The global stability of these control schemes is usually not guaranteed.

Another approach that is more rigorous, consists of explicitly introducing the

constraints in the lagrangian formulation of an optimal control problem (Mouradi

1979, Franckena and Sivan 1979, Gauthier and Bornard 1983). However, its

implementation isnot simple, because as an open-loop scheme it implies considerableoff-line computation and as a closed-loop scheme it is represented by a non-linear

controller.

The concept of invariance (or positive invariance), which is related to the notion of

Lyapunov functions, is a convenient tool both f or guaranteeing stability and

respecting the constraints. In the general case of constrained controllers for linear

systems, Gutman and Hagander (1985) used quadratic Lyapunov functions to

determine non-linear feedback controllers. However, for linear systems with linear

constraints on state and control variables, non-quadratic Lyapunov functions must be

used in order to generate the biggest positively invariant set included in the domain of

constraints. Such Lyapunov functions have already been applied for improving linearconstrained contr ollers of linear systems characterized by a stable non-negative

dynamic matrix (Chegan


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2. Problem statement

Throughout the paper, capital letters generally denote real matrices, lower case

letters denote column vectors or scalars, R n denotes the euclidean n-space and R n xm

the set of real n x m matrices. For a real matrix A = (aij), IAI denotes the matrix IAI =

(Ia i j !) ' For vectors x =[XI Xz ... Xn]T and Ixi =[ixil IXzl Ixnl]T. Finally:lL

denotes a unity matrix.We consider discrete-time linear system described by the difference equation

where X E R n, UE R m, A ER n XI', B E R n xm and k belongs to the set of non-negative

integers T= {a, 1,2, ... }.

The control vector u( k) is subject to constraints

where P = [PI pz ... P m ]T with P i> 0, i = 1,2, ... , m.

There is also given a bounded set of initial states Xodefined by the inequalities

whereGERqXnwithq~n,rankG=nandw=[wl Wz ... w q]Tw ith w i> O,i=

1,2, ..., q. These inequalities can also be considered as state constraints.

The problem to be studied is the determination of a linear state feedback control

law

that satisfy constraints (2) are transferred asymptotically to the origin while the

control vector u(k) does not violate the constraints (1). We call this problem the linear

constrained regulation problem (LCRP).

If the equilibrium x = of the open-loop system

is stable in the sense of Lyapunov or asymptotically stable, then the above problem

admits the trivial solution u(k) = 0. If, on the contrary, the open-loop system is

unstable, then the LCRP may not possess any solution. Therefore, we shall say that

constraints (1) and (2) are compatible with respect to system (S) if the LCRP has at

least one solution.

3. Existence conditions of linear constrained controllers

Let us associate to each linear state feedback control law u(k) = Fx(k) withF ERm x n the set

R(F, p ) = {x ER": - P~ Fx ~ p}

It is clear that the polyhedral set R(F, p ) isthe region of initial states of the closed-loopsystem (4) at which the linear state feedback control u(k) = Fx(k) does not initially

violate the constraints (1).

According to the above notation the set of initial states defined in (2) is expressed


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It is obvious that the control law u= Fx is a solution of the LCRP ifand only iftheresulting closed-loop system (4) is asymptotically stable and every trajectory x(k; xo)

emanating from the region R(G, w) does not leave the region R(F, p ) for any instant

k ET. This condition can also be expressed as follows (Bitsoris 1988 b).

Proposition 1

The control law u= Fx with FE Rm x n is a solution to the LCRP if and only if

(a) the eigenvalues Ai> i= 1,2, ... ,n , of the matrix A+ BF are in the open disklA d


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Proposition 2 (Bitsoris 1988 a)

The polyhedral set R( G, w) is a positively invariant set of system (S) if and only if

there exists a matrix H ERq x q such that

(IHI- :ll.)w :::; 0

GA-HG=O

By a direct application of this result to the closed-loop system described by (4) we

establish the following.

Proposition 3

If F ER '" x n and there exists a matrix HE R q x q such that

(IHI-:ll.)w:::;O

GA + GBF= HG

(iii) the eigenvalues Ai = 1,2, ...,n of the matrix A +B F are in the open disk IAi l


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Now, setting y= Gx in (10) and (11) we conclude that if

IGxl~w

or, equivalently ifx E R ( G, w) then

IF(GTG) -I GT

Gxl = IFxl ~ p

It must be noted that in the case where G ER " x n, inequality (9) becomes IFG-11

w ~ p and is,in addition, anecessary condition for R(G, w) c:; : R(F, p ) (Bitsoris 1988 b).

Now, by combining the results stated in Pr opositions 3 and 4 we conclude that if

FE R m x n and there exists an asymptotically stable matrix HE R q x q satisfying (6), (7)

and (9), then with the control law u= Fx all the states Xo ER(G, w) are transferredasymptotically to the origin while the control and state vectors satisfy inequalities (I)

and (2) respectively.

4. Design by means of linear programming

A straightforward application of the preceding result to the design of constrained

linear controllers seems to be a v ery difficult problem. However, by an appropriate

transformation of conditions (6), (7) and (9), the determination of a solution to the

LCRP can be reduced to a linear programming problem.

Observe that (7) is satisfied with

Now, by introducing matrix DE Rm

xq such that

D = F( GTG) - 1GT

relation (12) can be written as

H=GA(GTG)-IGT +GBD

Insertion of (13) and (14) into (9) and (6), respectively, yields

IDlw~p

IGA(GTG)-IGT +GBDlw~w

(15)

( 16)

These conditions on matrix D guarantee the existence of a linear control law

u= Fx = DGx such that R(G, w) is a positively invariant set of the resulting closed-loop system

and R(G, w) c :; : R(F, p) . However, conditions (15) and (16) do not imply the asympto-

tic stability of system (17). The asymptotic stability of (17) is guaranteed ifinequality

(16) is strictly satisfied. For this reason, inequality (16) is replaced by the inequality

IGA(GT G) -I GT + GBDlw ~ GW (18)

where G is a real number such that 0~ G


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Proposition 5

The matrix inequality

where Y = (Y iJ, Y ER P x r, r:J .E R' and fJ ER P with r:J .j ; ; : :'0 and f J i ; ;: :,0, is equivalent to the

set of equations

where e s = [ e 1S ez s e rsJT denotes one of the distinct vectors e s E R' with

components equal to +1or -1.

Proof

Assume that Y = (Y ij), Y E R Px r satisfies inequalities (19). Then

r r

r

IY ijejs r:J . j ~ I!Y ijllejs l r:J . j = IIY ij ! r:J . j ~ fJ ij~l j~l j~ l

for all i= 1,2, ... ,p and s= 1,2, ... ,2r, because r: J .j;;::' O.Therefore, (19) implies (20).

Conversely, if Y = (Y ij), Y ER P xr satisfies (20) then for ejS = sign (Y ij) we obtain

r r

I!Y ij! r:J . j= IY ijejs r:J . j ~ fJ i ' i= 1,2,... , pj~ 1 j~ 1

The application of this result to the determination of a solution to the LCRP isstraightforward. The system of inequalities (15) and (18) can be equivalently replaced

by a system of linear inequalities with unknown variables, the elements d ij of matrix D

and the positive variable B. If the set of solutions to these inequalities is non-empty,

then such a solution can be obtained by minimizing any linear function of the

unknown variables d ij and B.

Since it is very important not only to stabilize the system but also to increase the

rate of convergence to the equilibrium, we can choose as the objective function of the

linear programming problem, the function

J(D, B) = BIndeed, if B satisfy inequality (18), then by virtue of (14)

!H lw ~BW

v(x ) = max {!(G X );l}t Wi

is positive-definite. Therefore v(x) can be considered as the distance of x from the

origin. (It can easily be proved that v(x)

represents the distance of x from the origin, inthe space R" with the distance

{

V ( X ) + v(y)d(x, y) =

o

ifx#y

if x = Y


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Now, taking into account (7), we get

v(x(k + 1))= m~x f(GX(~~ 1))il}

= m~x {I(G (A + ~ :)X (k) );i}

= m~x fHG:;k))il}

~ m~x {(IHII~~(k)l)i}~ w(x(k))

because from (22) it follows that

Therefore, minimization of 8increases the rate of convergence of the state variable

x(k) to the origin.

If the optimal solution D*, 8* of the linear programming problem is such that

8*


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A~ [ _ ~ : ~~Jare Al = 1+jO-4 and .1 .2 = 1 - jO 4.Now, setting

B~[~ lG~ [-:5 ~ lw~[la p~7and taking into account that det G # - 0, conditions (15) and (18) become

IDlw":;p

IGA(GTG)-lGT + GBDlw= IGAG-I + GBDlw,,:;ew

51d11+ 1OId21 ,, : ; 7

510'87 +2 dll + 1010'58+ 2 d21 ,, : ; 5e

51-0'305 +2 dll + 1011'13+ 2d21, , : ; lO e

(26 a)

( 2 6 b )

(26 c)

Thus, the LCRP for system (23) under constraints (24) and (25) is reduced to the

determination of dl, d2 and e which minimize the objective function

under constraints (26).

Transformation of inequalities (26) to a system of linear inequalities and

application of a standard algorithm of linear programming gave the optimal values

u~ [-0435-04825{-:5 ~J

[::JWith this control, the resulting closed-loop system becomes x(k + 1)= (A

+ BD*G)x(k) where

[

08A+BD*G=

-0,11125

05 ]

-0,635

The eigenvalues of matrix A + BD*G are Al = 0'76, .1.2 = 059 and it is worth noticingthat the optimal value ofparameter e is a good upper bound of max ( 1.1.11 , 1.1 .21 ). Due tothe optimality of control law (26), the intersection of the boundary of set R ( G, w ) and

the boundary of set R(F, p) is not empty. This is shown in the following Figure.


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5. Conclusion

The Linear Constrained R egulation Problem has been analysed by determining

positively invariant sets associated to non-quadratic Lyapunov functions. Existence

conditions oflinear state feedback control laws have been obtained and translated into

an efficient algorithm based on linear programming. The control laws obtained by this

approach not only transfer to the origin all the initial states belonging to a polyhedral

subset of the state space but also optimize the convergence rate, while respecting

control constraints.

Although in the formulation of the LCRP no constraints on the state vector have

been imposed, the proposed algorithm also provides a solution to the case where the

state vector must satisfy linear inequalities.

REFERENCES

BITSORIS,G., 1986, Sur I'existence des ensembles invariants polyhedraux des systemes lineaires,

Technical Report 86015 (L.A.A.S.-CN.R.S, Toulouse, France); 1988 a, Int. J. Control,

47, 1713; 1988 b, J. Large-scale Systems, to be published.

BITSORIS,G., and BURGAT,C, 1977, Int. J. Control, 25, 413.

CHEGANc;AS,J., and BURGAT, C, 1985, Actes du Congres Automatique d'AFCET, Toulouse,

France, 193.

FRANKENA,J. F., and SIVAN, R., 1979, Int. J. Control, 30, 159.

GANTMACHER,F. R., 1960, The Theory of Matrices (New Yor k: Chelsea).

GAUTHIER, 1. P., and BORNARD, G., 1983, Rev. Autom. Inf. Res. Oper., 17, 205.GUTMAN,P.O., and HAGANDER, P., 1985, IEEE Trans. autom. Control, 30, 22.MOURADI,M., 1979, Rev. Autom. Inf. Res. Oper., 13, 127.

IJCFeedback control of linear discrete-time systems under state and control constraints88

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